In AABC, ZC = 90°. AD is the angular bisector of ZBAC meets BC at D. If CD=1.5cm and BD = 2.5cm, then find AC. Clear Response
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In given figure, check whether AD is the bisector of ∠ A of △ ABC in each of the following:
(i) AB = 5 cm, AC = 10 cm, BD = 1.5 cm and CD = 3.5 cm
(ii) AB = 4 cm, AC = 6 cm, BD = 1.6 cm and CD = 2.4 cm
(iii) AB = 8 cm, AC = 24 cm, BD = 6 cm and BC = 24 cm
(iv) AB = 6 cm, AC = 8 cm, BD = 1.5 cm and CD = 2 cm
(v) AB = 5 cm, AC = 12 cm, BD = 2.5 cm and BC = 9 cm
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Answer
If a line through one vertex of a triangle divides the opposite side in the ratio of the other two sides, then the line bisects the angle at the vertex.
That is if
AC
AB
=
DC
BD
then ∠BAD=∠CAD
1)
AB=5cm
AC=10cm
BD=1.5cm
CD=3.5cm
AC
AB
=
10
5
=0.5 (1)
CD
BD
=
3.5
1.5
≈0.43 (2)
(1)
=(2)
Hence AD is not the bisector of ∠A of △ ABC
2)
AB=4cm
AC=6cm
BD=1.6cm
CD=2.4cm
AC
AB
=
6
4
≈0.67 (1)
CD
BD
=
2.4
1.6
≈0.67 (2)
(1)=(2)
Hence AD is the bisector of ∠A of △ ABC
3)
AB=8cm
AC=24cm
BD=6cm
BC=24=BD+DC=6+DC
CD=18cm
AC
AB
=
24
8
≈0.33 (1)
CD
BD
=
18
6
≈0.33 (2)
(1)=(2)
Hence AD is the bisector of ∠A of △ ABC
4)
AB=6cm
AC=8cm
BD=1.5cm
CD=2cm
AC
AB
=
8
6
=0.75 (1)
CD
BD
=
2
1.5
=0.75 (2)
(1)=(2)
Hence AD is the bisector of ∠A of △ ABC
5)
AB=5cm
AC=12cm
BD=2.5cm
BC=9=BD+DC=2.5+DC
CD=6.5cm
AC
AB
=
12
5
≈0.42 (1)
CD
BD
=
6.5
2.5
≈0.38 (2)
(1)
=(2)
Hence AD is not the bisector of ∠A of △ ABC
Step-by-step explanation: